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nLab > Latest Changes: double gluing

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    • CommentRowNumber1.
    • CommentAuthorSam Staton
    • CommentTimeJul 29th 2019

    Page on double glueing

    v1, current

    • CommentRowNumber2.
    • CommentAuthorSam Staton
    • CommentTimeJul 29th 2019

    Reference

    v1, current

    • CommentRowNumber3.
    • CommentAuthorSam Staton
    • CommentTimeJul 29th 2019

    mention focuses

    v1, current

    • CommentRowNumber4.
    • CommentAuthorSam Staton
    • CommentTimeJul 30th 2019

    examples

    diff, v2, current

    • CommentRowNumber5.
    • CommentAuthorSam Staton
    • CommentTimeJul 30th 2019

    added example of quantum causal structure

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2019

    I finally got around to having a look at the Hyland-Schalk paper. In trying to make abstract sense of it, I realized that (unless I’m mistaken) their input datum for a double gluing construction — a lax symmetric monoidal functor L:C→EL:C\to E and a functor K:C→E opK:C\to E^{op} together with “contractions” L(R)⊗K(R⊗S)→K(S)L(R) \otimes K(R\otimes S) \to K(S) (section 4.2.1) — is just a lax symmetric monoidal functor C→Chu(E,1)C \to Chu(E,1). This explains their observation in section 4.3.1 that when CC is *\ast-autonomous any lax symmetric monoidal L:C→EL:C\to E extends canonically to a KK with contraction, by Pavlovic’s observation that ChuChu is right adjoint to the forgetful functor from *\ast-autonomous categories to closed symmetric monoidal categories equipped with a chosen object (and morphisms that colaxly preserve the object, which is automatic when the chosen object in the codomain is terminal).

    However, so far I haven’t been able to reformulate the actual construction of the double-glued category nicely in terms of this functor C→Chu(E,1)C\to Chu(E,1) as input, which is very unsatisfying. Any ideas?

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2019

    In particular, the ordinary gluing of (L,K):C→Chu(E,1)(L,K) : C\to Chu(E,1) would have, as objects, tuples consisting of an object R∈CR\in C, an object (U,X)∈Chu(E,1)(U,X)\in Chu(E,1) — which is to say two objects U,X∈EU,X\in E — and a morphism (U,X)→(L,K)(R)(U,X) \to (L,K)(R) in Chu(E,1)Chu(E,1) — which is to say morphisms U→L(R)U\to L(R) and K(R)→XK(R) \to X in EE. This looks almost like the double gluing except that the morphism K(R)→XK(R) \to X goes the wrong way! Unless I am confused somehow.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2019
    • (edited Oct 13th 2019)

    Ah, I got it! It’s another comma double category. Regarding CC and Chu(E,1)Chu(E,1) as vertically discrete double categories, we can map them both into the double Chu construction ℂhu(E,1)\mathbb{C}hu(E,1) (the inclusion of Chu(E,1)Chu(E,1) being an isomorphism onto the horizontal category). Then the double gluing category is the horizontal category of the comma object in double categories (and vertical transformations) of the cospan C→ℂhu(E,1)←Chu(E,1)C \to \mathbb{C}hu(E,1) \leftarrow Chu(E,1). And to encode the monoidal structure, we can use double polycategories (or multicategories) instead.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2019

    Added the general case, in terms of comma double polycategories.

    diff, v5, current

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